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Wikipedia / Moment generating function

In probability theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification of the random variable's probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment generating functions.

As its name implies, the moment generating function can be used to compute a distribution’s moments: the -th moment about 0 is the -th derivative of the moment generating function, evaluated at 0.

In addition to univariate real-valued distributions, moment generating functions can also be defined for vector- or matrix-valued random variables, and can even be extended to more general cases.

The moment generating function of a real-valued distribution does not always exist, unlike the characteristic function. There are relations between the behavior of the moment generating function of a distribution and properties of the distribution, such as the existence of moments.

Definition

be a random variable with CDF, is

provided this expectation in some open neighborhood exists. If the expectation does not exist in an open neighborhood of 0, we say that the moment generating function does not exist.

In other words, the moment generating function of is the expectation-dimensional random vector instead of

always exists and is equal to1. However, a key problem with moment generating functions is that moments and the moment generating function may not exist, as the integrals need not converge absolutely. By contrast, the characteristic function or Fourier transform always exists (because it is the integral of a bounded function on a space of finite measure), and for some purposes may be used instead.

The moment generating function is so named because it can be used to find the moments of the distribution. is

is the moment.

and the two-sided Laplace transform holds:

since the PDF's two-sided Laplace transform is given as

and the moment generating function's definition expands (by the law of the unconscious statistician

being a Wick rotation is the Fourier transform is of exponential order is a Wick rotation of its two-sided Laplace transform in the region of convergence. See the relation of the Fourier and Laplace transforms for further information.

Examples

Here are some examples of the moment generating function and the characteristic function for comparison. It can be seen that the characteristic function is a Wick rotation when the latter exists.

Calculation

The moment generating function is the expectation of a function of the random variable, it can be written as:

has a continuous probability density function is the two-sided Laplace transform.

th moment.

Linear transformations of random variables

Linear combination of independent random variables

, where the are independent random variables and the are constants, then the probability density function for is the convolution of the probability density functions of each of the , and the moment generating function for is given by

Vector-valued random variables

For vector-valued random variables with real components, the moment generating function is given by

is the dot product.

Important properties

Moment generating functions are positive and log-convex, with M(0) = 1.

are two random variables and for all values of,

for all values of (or equivalently and have the same distribution). This statement is not equivalent to the statement "if two distributions have the same moments, then they are identical at all points." This is because in some cases, the moments exist and yet the moment generating function does not, because the limit

may not exist. The log-normal distribution is not defined for any positive as the integral diverges and so not in a neighbourhood of 0; there are other distributions with the same moments.

Calculations of moments

The moment generating function is so called because if it exists on an open interval around , then it is the exponential generating function of the moments of the probability distribution:

That is, with being a nonnegative integer, the -th moment about 0 is the -th derivative of the moment generating function, evaluated at .

Other properties

Jensen's inequality is the mean of .

The moment generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable . This statement is also called the Chernoff bound and any exists. For example, when , which is within a factor of of the exact value.

Various lemmas, such as Hoeffding's lemma or Bennett's inequality provide bounds on the moment generating function in the case of a zero-mean, bounded random variable.

.

for any .

degrees of freedom. Then from the examples We know that in this case. The moment generating function bound is thus very strong in this case.

Relation to other functions

Related to the moment generating function are a number of other transforms that are common in probability theory:

  1. Characteristic function: The characteristic function the characteristic function is the moment generating function of iX or the moment generating function of X evaluated on the imaginary axis. This function can also be viewed as the Fourier transform of the probability density function, which can therefore be deduced from it by inverse Fourier transform.

  2. Cumulant-generating function: The cumulant-generating function is defined as the logarithm of the moment generating function; some instead define the cumulant-generating function as the logarithm of the characteristic function, while others call this latter the second cumulant-generating function.

  3. Probability-generating function: The probability-generating function

See also

References

Citations

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